Gauge-invariant coherent structures and modulational instability in a stochastic Yajima–Oikawa system

This paper investigates a stochastic Yajima–Oikawa long-wave–short-wave resonance system with multiplicative Stratonovich phase noise acting on the short-wave component. By embedding the Brownian motion into the carrier phase, we show that the stochastic perturbation is gauge-removable from the amplitude dynamics. Consequently, the traveling-wave reduction leads to a deterministic Duffing-type profile equation while the long-wave component is slaved to the short-wave intensity through an explicit algebraic relation. Using the associated Hamiltonian structure, we classify the admissible dynamical regimes and derive explicit solitary-wave solutions, kink-type amplitude waves with dark-notch intensity profiles on a nonzero background, and Jacobi elliptic periodic waves. The homoclinic, heteroclinic, and closed periodic trajectories of the reduced phase plane are directly connected with these wave families. We also perform a modulational-instability analysis of the continuous-wave background and prove that, for the present pure Stratonovich phase-noise model, the instability threshold and growth law coincide with those of the gauge-equivalent deterministic system. Finally, we derive phase-sensitive stochastic observables showing that Brownian phase noise produces exponential attenuation of the ensemble mean field and temporal correlation decay, whereas the short-wave intensity and the induced long-wave response remain deterministic. The results clarify how random carrier-phase fluctuations influence observable phase coherence without changing the leading-order coherent envelope mechanism.